\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Introduction to ray tracing " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Ray tracing is a tool used in many physics discipline since high frequency waves can accurately be approximated by rays. Typical applications include 3D rendering (think [povray](http://www.povray.org)), lens design or acoustic wave simulation (which is what I do professionally).\n", "To trace rays, you usually start with a source and follow reflexions and refractions until some the end of the tracing (exiting the scene, arriving at the camera).\n", "\n", "In this context, the goal for this post is to find the ray that connects the source to the camera through a set of reflexions. Let's define the reference configuration for this post. We'll model points with Numpy arrays of $(x, y)$ coordinates. \n", "\n", "We define a source and a camera, as well as an interface that separates the two bounding media that have different speeds." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "source = np.array([0., 1.]).reshape(-1, 1)\n", "camera = np.array([1., -1.]).reshape(-1, 1)\n", "interface_y = 0." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's plot this." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "%matplotlib notebook" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " '');\n", " var titletext = $(\n", " '');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('');\n", " button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n", " 'ui-button-icon-only');\n", " button.attr('role', 'button');\n", " button.attr('aria-disabled', 'false');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", "\n", " var icon_img = $('');\n", " icon_img.addClass('ui-button-icon-primary ui-icon');\n", " icon_img.addClass(image);\n", " icon_img.addClass('ui-corner-all');\n", "\n", " var tooltip_span = $('');\n", " tooltip_span.addClass('ui-button-text');\n", " tooltip_span.html(tooltip);\n", "\n", " button.append(icon_img);\n", " button.append(tooltip_span);\n", "\n", " nav_element.append(button);\n", " }\n", "\n", " var fmt_picker_span = $('');\n", "\n", " var fmt_picker = $('');\n", " fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n", " fmt_picker_span.append(fmt_picker);\n", " nav_element.append(fmt_picker_span);\n", " this.format_dropdown = fmt_picker[0];\n", "\n", " for (var ind in mpl.extensions) {\n", " var fmt = mpl.extensions[ind];\n", " var option = $(\n", " '', {selected: fmt === mpl.default_extension}).html(fmt);\n", " fmt_picker.append(option)\n", " }\n", "\n", " // Add hover states to the ui-buttons\n", " $( \".ui-button\" ).hover(\n", " function() { $(this).addClass(\"ui-state-hover\");},\n", " function() { $(this).removeClass(\"ui-state-hover\");}\n", " );\n", "\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "}\n", "\n", "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n", " // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n", " // which will in turn request a refresh of the image.\n", " this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n", "}\n", "\n", "mpl.figure.prototype.send_message = function(type, properties) {\n", " properties['type'] = type;\n", " properties['figure_id'] = this.id;\n", " this.ws.send(JSON.stringify(properties));\n", "}\n", "\n", "mpl.figure.prototype.send_draw_message = function() {\n", " if (!this.waiting) {\n", " this.waiting = true;\n", " this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n", " }\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " var format_dropdown = fig.format_dropdown;\n", " var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n", " fig.ondownload(fig, format);\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_resize = function(fig, msg) {\n", " var size = msg['size'];\n", " if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n", " fig._resize_canvas(size[0], size[1]);\n", " fig.send_message(\"refresh\", {});\n", " };\n", "}\n", "\n", "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n", " var x0 = msg['x0'] / mpl.ratio;\n", " var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n", " var x1 = msg['x1'] / mpl.ratio;\n", " var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n", " x0 = Math.floor(x0) + 0.5;\n", " y0 = Math.floor(y0) + 0.5;\n", " x1 = Math.floor(x1) + 0.5;\n", " y1 = Math.floor(y1) + 0.5;\n", " var min_x = Math.min(x0, x1);\n", " var min_y = Math.min(y0, y1);\n", " var width = Math.abs(x1 - x0);\n", " var height = Math.abs(y1 - y0);\n", "\n", " fig.rubberband_context.clearRect(\n", " 0, 0, fig.canvas.width, fig.canvas.height);\n", "\n", " fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n", "}\n", "\n", "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n", " // Updates the figure title.\n", " fig.header.textContent = msg['label'];\n", "}\n", "\n", "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n", " var cursor = msg['cursor'];\n", " switch(cursor)\n", " {\n", " case 0:\n", " cursor = 'pointer';\n", " break;\n", " case 1:\n", " cursor = 'default';\n", " break;\n", " case 2:\n", " cursor = 'crosshair';\n", " break;\n", " case 3:\n", " cursor = 'move';\n", " break;\n", " }\n", " fig.rubberband_canvas.style.cursor = cursor;\n", "}\n", "\n", "mpl.figure.prototype.handle_message = function(fig, msg) {\n", " fig.message.textContent = msg['message'];\n", "}\n", "\n", "mpl.figure.prototype.handle_draw = function(fig, msg) {\n", " // Request the server to send over a new figure.\n", " fig.send_draw_message();\n", "}\n", "\n", "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n", " fig.image_mode = msg['mode'];\n", "}\n", "\n", "mpl.figure.prototype.updated_canvas_event = function() {\n", " // Called whenever the canvas gets updated.\n", " this.send_message(\"ack\", {});\n", "}\n", "\n", "// A function to construct a web socket function for onmessage handling.\n", "// Called in the figure constructor.\n", "mpl.figure.prototype._make_on_message_function = function(fig) {\n", " return function socket_on_message(evt) {\n", " if (evt.data instanceof Blob) {\n", " /* FIXME: We get \"Resource interpreted as Image but\n", " * transferred with MIME type text/plain:\" errors on\n", " * Chrome. But how to set the MIME type? It doesn't seem\n", " * to be part of the websocket stream */\n", " evt.data.type = \"image/png\";\n", "\n", " /* Free the memory for the previous frames */\n", " if (fig.imageObj.src) {\n", " (window.URL || window.webkitURL).revokeObjectURL(\n", " fig.imageObj.src);\n", " }\n", "\n", " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", "\n", " var msg = JSON.parse(evt.data);\n", " var msg_type = msg['type'];\n", "\n", " // Call the \"handle_{type}\" callback, which takes\n", " // the figure and JSON message as its only arguments.\n", " try {\n", " var callback = fig[\"handle_\" + msg_type];\n", " } catch (e) {\n", " console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n", " return;\n", " }\n", "\n", " if (callback) {\n", " try {\n", " // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n", " callback(fig, msg);\n", " } catch (e) {\n", " console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n", " }\n", " }\n", " };\n", "}\n", "\n", "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n", "mpl.findpos = function(e) {\n", " //this section is from http://www.quirksmode.org/js/events_properties.html\n", " var targ;\n", " if (!e)\n", " e = window.event;\n", " if (e.target)\n", " targ = e.target;\n", " else if (e.srcElement)\n", " targ = e.srcElement;\n", " if (targ.nodeType == 3) // defeat Safari bug\n", " targ = targ.parentNode;\n", "\n", " // jQuery normalizes the pageX and pageY\n", " // pageX,Y are the mouse positions relative to the document\n", " // offset() returns the position of the element relative to the document\n", " var x = e.pageX - $(targ).offset().left;\n", " var y = e.pageY - $(targ).offset().top;\n", "\n", " return {\"x\": x, \"y\": y};\n", "};\n", "\n", "/*\n", " * return a copy of an object with only non-object keys\n", " * we need this to avoid circular references\n", " * http://stackoverflow.com/a/24161582/3208463\n", " */\n", "function simpleKeys (original) {\n", " return Object.keys(original).reduce(function (obj, key) {\n", " if (typeof original[key] !== 'object')\n", " obj[key] = original[key]\n", " return obj;\n", " }, {});\n", "}\n", "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", " if (name === 'button_press')\n", " {\n", " this.canvas.focus();\n", " this.canvas_div.focus();\n", " }\n", "\n", " var x = canvas_pos.x * mpl.ratio;\n", " var y = canvas_pos.y * mpl.ratio;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", " step: event.step,\n", " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", " * to control all of the cursor setting manually through the\n", " * 'cursor' event from matplotlib */\n", " event.preventDefault();\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " // Handle any extra behaviour associated with a key event\n", "}\n", "\n", "mpl.figure.prototype.key_event = function(event, name) {\n", "\n", " // Prevent repeat events\n", " if (name == 'key_press')\n", " {\n", " if (event.which === this._key)\n", " return;\n", " else\n", " this._key = event.which;\n", " }\n", " if (name == 'key_release')\n", " this._key = null;\n", "\n", " var value = '';\n", " if (event.ctrlKey && event.which != 17)\n", " value += \"ctrl+\";\n", " if (event.altKey && event.which != 18)\n", " value += \"alt+\";\n", " if (event.shiftKey && event.which != 16)\n", " value += \"shift+\";\n", "\n", " value += 'k';\n", " value += event.which.toString();\n", "\n", " this._key_event_extra(event, name);\n", "\n", " this.send_message(name, {key: value,\n", " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n", " if (name == 'download') {\n", " this.handle_save(this, null);\n", " } else {\n", " this.send_message(\"toolbar_button\", {name: name});\n", " }\n", "};\n", "\n", "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n", " this.message.textContent = tooltip;\n", "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", " // object with the appropriate methods. Currently this is a non binary\n", " // socket, so there is still some room for performance tuning.\n", " var ws = {};\n", "\n", " ws.close = function() {\n", " comm.close()\n", " };\n", " ws.send = function(m) {\n", " //console.log('sending', m);\n", " comm.send(m);\n", " };\n", " // Register the callback with on_msg.\n", " comm.on_msg(function(msg) {\n", " //console.log('receiving', msg['content']['data'], msg);\n", " // Pass the mpl event to the overriden (by mpl) onmessage function.\n", " ws.onmessage(msg['content']['data'])\n", " });\n", " return ws;\n", "}\n", "\n", "mpl.mpl_figure_comm = function(comm, msg) {\n", " // This is the function which gets called when the mpl process\n", " // starts-up an IPython Comm through the \"matplotlib\" channel.\n", "\n", " var id = msg.content.data.id;\n", " // Get hold of the div created by the display call when the Comm\n", " // socket was opened in Python.\n", " var element = $(\"#\" + id);\n", " var ws_proxy = comm_websocket_adapter(comm)\n", "\n", " function ondownload(figure, format) {\n", " window.open(figure.imageObj.src);\n", " }\n", "\n", " var fig = new mpl.figure(id, ws_proxy,\n", " ondownload,\n", " element.get(0));\n", "\n", " // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n", " // web socket which is closed, not our websocket->open comm proxy.\n", " ws_proxy.onopen();\n", "\n", " fig.parent_element = element.get(0);\n", " fig.cell_info = mpl.find_output_cell(\"\");\n", " if (!fig.cell_info) {\n", " console.error(\"Failed to find cell for figure\", id, fig);\n", " return;\n", " }\n", "\n", " var output_index = fig.cell_info[2]\n", " var cell = fig.cell_info[0];\n", "\n", "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", " var width = fig.canvas.width/mpl.ratio\n", " fig.root.unbind('remove')\n", "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", " // Re-enable the keyboard manager in IPython - without this line, in FF,\n", " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('Have been enjoying autograd to get derivatives automatically (to plug into scipy.optimize). Think in 2018, there's no reason to compute gradients by hand anymore: https://t.co/52P1pvQjpy
— Erik Bernhardsson (@fulhack) January 26, 2018